The classification of normalizing groups

Let X be a finite set such that vertical bar X vertical bar = n. Let T-n and S-n denote the transformation monoid and the symmetric group on n points, respectively. Given a is an element of T-n \ S-n, we say that a group G <= S-n is a-normalizing if < a, G > \ G = < g(-1)ag vertical bar g is an element of G >, where < a, G > and < g(-1)ag vertical bar g is an element of G > denote the subsemigroups of T-n generated by the sets {a} boolean OR G and {g(-1)ag vertical bar g is an element of G}, respectively. If G is a-normalizing for all a is an element of T-n \ S-n, then we say that G is normalizing. The goal of this paper is to classify the normalizing groups and hence answer a question of Levi, McAlister, and McFadden. The paper ends with a number of problems for experts in groups, semigroups and matrix theory. (C) 2012 Elsevier Inc. All rights reserved.